Four balances are shown where the two sides of each contain exactly the same amount of weight. The first balance has 4 squares, 1 triangle and 1 ball on the left side, right side has 2 diamonds. The second balance has 1 triangle and 1 square on the left side, 1 ball on the right side. The third balance has 1 triangle and 1 diamond on the left side, 1 ball and 2 squares on the right side. The fourth triangle has 1 diamond, 1 triangle and 1 square on the left side. How many balls are on the right side? Explain/show how this was determined.
I got 3B on the last scale
balance 1: 4S+1T+1B=2D
balance 2: 1S+1T=1B
banace 3: 1T+1D=1B+2S
balance 4: 1S+1T+1D= XXXX B
solve for XXXX
now, rearranging the four equations in order of BDST
a) 1B-2D+4S+T=0
b) -1B+0D+1S+T=0
c) -1B+D-2S+1T=0
d) XXXB+D+S+T=0
subtract d from b
(1-xx)B+D=0
add a + C
-D+2S+2T=0
now add these two equations
(1-xx)B+2S+2T=0 , now subtract 2*b from this
(xx-1+2)B =0
which means xxx=1
check my work on this, it is easy to make an error.
To determine the number of balls on the right side, we need to analyze the information given for each balance step-by-step.
Balance 1:
Left side: 4 squares, 1 triangle, and 1 ball.
Right side: 2 diamonds.
Since the two sides of the balance are equal, we can infer that 4 squares + 1 triangle + 1 ball is equal to 2 diamonds. However, the number of balls is not mentioned in this balance.
Balance 2:
Left side: 1 triangle and 1 square.
Right side: 1 ball.
In this balance, we can see that 1 triangle + 1 square is equal to 1 ball.
Balance 3:
Left side: 1 triangle and 1 diamond.
Right side: 1 ball and 2 squares.
Here, we can see that 1 triangle + 1 diamond is equal to 1 ball + 2 squares. However, the number of balls is still unknown.
Balance 4:
Left side: 1 diamond, 1 triangle, and 1 square.
In this balance, we can see that 1 diamond + 1 triangle + 1 square is equal to an unknown weight on the right side.
Now, by combining the information from all balances, we can solve for the number of balls on the right side.
From Balance 1, we know that the weight of 4 squares + 1 triangle + 1 ball is equal to the weight of 2 diamonds. However, the number of balls is not known.
From Balance 2, we know that 1 triangle + 1 square is equal to 1 ball.
From Balance 3, we know that 1 triangle + 1 diamond is equal to 1 ball + 2 squares. However, the number of balls is still unknown.
From Balance 4, we know that 1 diamond + 1 triangle + 1 square is equal to an unknown weight.
By comparing the information, we can conclude that 1 ball is equal to 1 triangle + 1 square. Therefore, in Balance 4, where the left side has 1 diamond, 1 triangle, and 1 square, the weight on the right side would be 1 ball.
Thus, the number of balls on the right side is 1.
To determine how many balls are on the right side, we need to analyze each balance individually and compare the weights on both sides.
1) First balance:
- Left side: 4 squares + 1 triangle + 1 ball
- Right side: 2 diamonds
Since both sides are balanced, the weight on the left side must be equal to the weight on the right side. The left side has 1 ball, so the right side must have 1 ball as well.
2) Second balance:
- Left side: 1 triangle + 1 square
- Right side: 1 ball
Again, both sides are balanced, so the weight on the left side is equal to the weight on the right side. The left side has 1 ball, so the right side must have 1 ball as well.
3) Third balance:
- Left side: 1 triangle + 1 diamond
- Right side: 1 ball + 2 squares
Once again, both sides are balanced. The left side does not have any balls, but the right side does. So, the right side has at least 1 ball.
4) Fourth balance:
- Left side: 1 diamond + 1 triangle + 1 square
We don't know the weight on the right side, but since the left side does not have a ball, we can conclude that the number of balls on the right side is still at least 1.
In summary, after analyzing all four balances, we can determine that there are at least 1 ball on the right side.